KED justification

Intuitively, the safety of a dart depends on the dart's mass, the
dart's muzzle velocity (the maximum speed the dart will move at unless
it is shot down a tall cliff or something similar), the minimum impact
area, the hardness of the dart tip, and the thickness of the dart tip.
Heavier darts are less safe; they are known to hurt more. Faster darts
are known to hurt more. Intuitively, darts with pointy tips are known to
hurt more and can readily penetrate flesh. Harder tips hurt more than
soft tips.

The KED formula fits out intuition except with regard to tip
softness. KED says nothing about the softness of the tip. A different
dynamic analysis must be used in that case. This analysis will be
detailed later. Note that a class in mechanical vibrations will provide
more than enough information to characterize soft tips.

Typical KED values

The table below is based on data from Klaviel posted by Daniel Beaver at NerfHaven[1][2]. Slug darts were used; a dart mass of 0.9 g was assumed in the KED calculations (#6 washers were used in the slugs).

<math>V_m \left(\tfrac{\text{ft}}{\text{s}}\right)\,\!</math>

<math>E_k^{} \left(\tfrac{\text{mJ}}{\text{mm}^2}\right)\,\!</math>

3B (k26 and perfect seal)

175

9.0

SNAP 5 with hopper

190

11

SNAP 2 with hopper

200

12

PAC with hopper

200

12

Ryan's singled +bow

210

13

The U3's SM 1500

250

18

Ice9's Big Blast

350

36

KED limits

A critical KED (i.e. one that will start to damage a surface) is given the notation <math>E_{crit}^{}\,\!</math>.

The values for eye, skin, and bone damage below should be thought
of as good estimates of the average values. Much more testing is
necessary to determine the average value and distribution of the KED
necessary to penetrate the body. This testing is gruesome, as it
generally involves shooting dead rabbits or pigs in the eyes, and future
testing is not likely to be performed for this reason. Please use
generous safety factors. This information is provided to put the power
of a Nerf gun into perspective so that this power may be fully
respected.

A note for the squeamish

Most of the references containing critical KED data also have
photographs of gunshot wounds that some readers may find to be
disturbing. Don't look at these references if you are squeamish.

Hasbro (SRS-045)

Hasbro's SRS-045[3]
sets a KED limit of 1.6
<math>\tfrac{\text{mJ}}{\text{mm}^2}</math> for projectiles
with a KE over 80 mJ. This limitation is based on ISO #8124[4]. Hasbro usually designs for 20% under this limit to account for manufacturing variations[5].

Eye damage

Some data for spheres is available. Admittedly, spheres are not
particularly similar to Nerf darts, but many darts have domed heads, so
the data should be correct in that case, and approximately correct for
flat headed darts.

For a summary and analysis of data available up until 1994, see Sellier and Kneubuehl[6].

This data suggests there is a relationship between critical KED
and projectile diameter. However, the data for diameters 3.2 mm and
higher (most Nerf darts have a diameter of about 12.7 mm) is fairly
consistent, and that data has an average value of <math>E_{crit}^{}\,\!</math> of 62.0 <math>\tfrac{\text{mJ}}{\text{mm}^2}\,\!</math>.

This data is for penetrating the eye, not simply causing damage.
The eye can be damaged at levels below the listed critical KEDs.

Damage to material (linear elastic case)

The material can be assumed to deformed in a simple manner as a
simplification: the projection of the impact area deforms like a linear
spring and nothing else deforms.

How much energy per unit volume can a material absorb before
permanently deforming? This energy is called
<math>U_r\,\!</math>, the modulus of resilence.
The answer can be found rather easily with Hooke's law and the
work-energy theorem (normalized per unit volume).
<math>\sigma_y\,\!</math> is the yield stress and <math>\varepsilon_y\,\!</math> is the yield strain,

This material property can be multiplied by the thickness of the
material to find a critical KED value for that material for yielding.

<math>E_{crit}^{} = U_r t = \frac{t \sigma_y^2}{2 E}\,\!</math>

Note that yielding really only is denting; if the KED value is
less than the critical KED value, the material will completely defeat
the projectile without any denting (unless fatigue is considered). This
information can be directly used to determine whether eyewear will
protect against a dart.

If the KED value is greater than the critical KED value, all that
can be said is that at least the material will be slightly dented.
Potentially the projectile could dent or fracture the target and bounce
off. Potentially the projectile could be embedded in the target.
Potentially the projectile could pass completely through the sheet. This
analysis offers little indication about which is more likely. Sellier
and Kneubuehl[6] have some information about projectiles moving through objects they impact.

Different critical KED values for complete defeat of the material must be found experimentally or via finite element analysis.

Measuring KED

Chronograph, scale, and calipers

A chronograph,
a small scale, and potentially some calipers can be used to find KED.
Measure the muzzle velocity of the blaster, the mass of the dart, and
the diameter of the dart. These numbers can be directly plugged into the
KED equation.

Ballistic pendulum

A ballistic pendulum can be used to directly find the kinetic energy
of the dart, and from there the KED can be found by dividing the KE by
the minimum area. A ballistic pendulum is far easier to build than a
homemade chronograph, as it only involves a pendulum and some weighted
target for the projectile. A styrofoam target should work nicely for
Nerf applications.

Material penetration

Using the information in the "Damage to material" section above, one
can figure out what thickness of a certain material is necessary to dent
it. Practically speaking, however, this is not useful as denting any
material a Nerf gun could easily shoot through like cardboard is very
simple.

Empirical testing with a sheet of cardboard or something similar
can find the KED necessary to penetrate the sheet, and this sheet can
then be used to test whether the KED of a blaster is below or above a
certain limit.

Other notations

Sellier and Kneubuehl[6]
use <math>E_{ths}^'\,\!</math> for the critical kinetic
energy density. "Ths" refers to a threshold; this is the threshold to
start penetrating something. This notation's use of a prime does not
make sense as the primes in this context generally denote normalization
with respect to one length. Two primes are necessary for normalization
with respect to an area (i.e. a length squared).

↑ 6.06.16.2 Karl G. Sellier and Beat P. Kneubuehl, Wound ballistics and the scientific background (Elsevier Health Sciences, 1994). http://books.google.com/books?id=jZf1GaXQUvQC
(Note: This book is riddled with mistakes. Do some checks for
consistency whenever taking data from a table. Also check to see if the
right names on papers were used. The authors used Stuart when Stewart
was the correct name on one paper they cite, for example.)

↑ Kramer D Powley et al.,
“Velocity necessary for a BB to penetrate the eye: an experimental
study using pig eyes,” The American Journal of Forensic Medicine and
Pathology: Official Publication of the National Association of Medical
Examiners 25, no. 4 (December 2004): 273-275.